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Quantum Field Theory and Critical Phenomena$
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Jean Zinn-Justin

Print publication date: 2002

Print ISBN-13: 9780198509233

Published to Oxford Scholarship Online: January 2010

DOI: 10.1093/acprof:oso/9780198509233.001.0001

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The O(N) Vector Model For N Large

The O(N) Vector Model For N Large

Quantum Field Theory and Critical Phenomena


Oxford University Press

The preceding chapters derived universal properties of critical systems within the framework of the formal ε = 4 - d expansion. It is, therefore, reassuring to verify that the results obtained in this way remain valid, at least in some limiting case, even when ε is no longer in infinitesimal. This chapter shows that in the case of the O(N) symmetric (φ2)2 field theory, the same universal properties can also be derived at fixed dimension in the large N limit, and more generally order by order in an 1/N-expansion. Large N techniques are also useful because they the discussion of other non-perturbative questions, including issues relevant to four-dimensional physics like renormalons and triviality. Finally, the chapter exhibits a surprising relation between the (φ2)2 field theory and the nonlinear σ-model valid to all orders in the 1/N expansion.

Keywords:   four-dimensional physics, N limit, non-perturbative questions, field theory

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